Finite Group Databasebeta

← #34180913

Subgroups — #34180913

Order 20224.

778 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579152C211yesmaximal, characteristic
2#23763478C212yesmaximal
3#23761730C211yesmaximal, characteristic
4#23761729C212yesmaximal
5C79×#1463C211yesmaximal, characteristic
6C2×#14579152—24no—
7C2×#14579152—24no—
8C2×#14579152—22no—
9C2×#14579152—22no—
10C22×C4⋊Dic79C411yescharacteristic
11C22×C4⋊Dic79C411yescharacteristic
12C22×C4⋊Dic79C2211yescharacteristic
13C22×Dic79⋊C4C411yescharacteristic
14C22×Dic79⋊C4C2211yescharacteristic
15C22×C4×Dic79C411yescharacteristic
16#19098527C2212yes—
17#19098331—24no—
18#19098289—22no—
19#19098288—24no—
20C22×C4⋊C4×C79C2211yescharacteristic
21#407×C79C2212yes—
22#324×C79—22no—
23C23×Dic79C2×C411yescharacteristic
24C23×Dic79C2×C411yescharacteristic
25C23×Dic79C2×C411yescharacteristic
26C23×Dic79C2×C411yescharacteristic
27C2×#11929608—22no—
28#14579152—216no—
29#14579152—216no—
30C2×C4⋊Dic79—22no—
31C2×C4⋊Dic79—22no—
32C2×C4⋊Dic79—22no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79—22no—
36C2×C4⋊Dic79—22no—
37C2×C4⋊Dic79—22no—
38C2×C4⋊Dic79—22no—
39C2×C4⋊Dic79D414yes—
40C2×Dic79⋊C4—22no—
41C2×Dic79⋊C4—22no—
42C2×Dic79⋊C4—21no—
43C2×Dic79⋊C4—21no—
44C2×Dic79⋊C4—22no—
45C2×Dic79⋊C4—22no—
46C2×Dic79⋊C4D414yes—
47C2×C4×Dic79—22no—
48C2×C4×Dic79—22no—
49C2×C4×Dic79—22no—
50C22⋊(C79⋊C8)—22no—
51C22⋊(C79⋊C8)—22no—
52C23×C4×C79C2×C411yescharacteristic
53C23×C4×C79C2×C411yescharacteristic
54C23×C4×C79C2311yescharacteristic
55C2×M16×C79—22no—
56C2×C4⋊C4×C79—22no—
57C2×C4⋊C4×C79—22no—
58C2×C4⋊C4×C79D412yes—
59C2×C4⋊C4×C79D412yes—
60C22⋊C8×C79—22no—
61C22⋊C8×C79—22no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—42no—
71C22×Dic79—42no—
72C22×Dic79—42no—
73C22×Dic79—42no—
74C22×Dic79C22⋊C412yes—
75C22×Dic79C22⋊C412yes—
76C22×Dic79C22⋊C412yes—
77C22×Dic79C22⋊C412yes—
78#11929608—44no—
79C2×C79⋊C8—42no—
80C4⋊Dic79—28no—
81C4⋊Dic79—28no—
82C4⋊Dic79—24no—
83C4⋊Dic79—24no—
84Dic79⋊C4—28no—
85Dic79⋊C4—24no—
86Dic79⋊C4—24no—
87C4×Dic79—28no—
88C24×C79C22×C411yescharacteristic
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—24no—
92C22×C4×C79—21no—
93C22×C4×C79—21no—
94C22×C4×C79—42no—
95C22×C4×C79—42no—
96C22×C4×C79C22⋊C412yes—
97C22×C4×C79C22⋊C412yes—
98C22×C4×C79C2×D412yes—
99C22×C4×C79C2×D412yes—
100C22×C4×C79C2×D411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102M16×C79—44no—
103C2×C8×C79—42no—
104C4⋊C4×C79—24no—
105C4⋊C4×C79—24no—
106C23×C79—22no—
107C23×C79—22no—
108C23×C79—22no—
109C23×C79C2×C22⋊C411yescharacteristic
110C23×C79C2×C22⋊C411yescharacteristic
111C23×C79C2×C22⋊C411yescharacteristic
112C2×C4×C79—24no—
113C2×C4×C79—24no—
114C2×C4×C79—22no—
115C2×C4×C79—24no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—42no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79#10114yes—
125C2×C4×C79#12012yes—
126C2×C4×C79#12012yes—
127C2×Dic79—22no—
128C2×Dic79—22no—
129C2×Dic79—22no—
130C2×Dic79—22no—
131C2×Dic79—24no—
132C2×Dic79—42no—
133C2×Dic79—42no—
134C2×Dic79—42no—
135C2×Dic79—42no—
136C2×Dic79—42no—
137C2×Dic79—42no—
138C2×Dic79—42no—
139C2×Dic79—42no—
140C2×Dic79—42no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C8×C79—82no—
144C79⋊C8—82no—
145C22×C79—28no—
146C22×C79—21no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79#37911yescharacteristic
153C22×C79#41012yes—
154C22×C79C2×#10112yes—
155C22×C79#41312yes—
156C4×C79—22no—
157C4×C79—22no—
158C4×C79—24no—
159C4×C79—42no—
160Dic79—42no—
161Dic79—42no—
162Dic79—42no—
163Dic79—42no—
164#8554—791nomaximal, Sylow
165C2×C79—22no—
166C2×C79—22no—
167C2×C79—22no—
168C2×C79#178211yescharacteristic
169C2×C79#178411yescharacteristic
170C2×C79#178511yescharacteristic
171C22×#125—791no—
172#1821—792no—
173#1463—792no—
174#1463—791no—
175#1463D7911yescharacteristic
176C79#855411yescharacteristic, complemented, Sylow
177C2×#125—1584no—
178C2×#125—1584no—
179C2×#125—1582no—
180C2×#125—1582no—
181C22×C4⋊C4—791no—
182C22×C4⋊C4—791no—
183C22×C4⋊C4—791no—
184C22×C4⋊C4—791no—
185C22×C4⋊C4—791no—
186C22×C4⋊C4C2×D7911yescharacteristic
187C22×C42—791no—
188#407—792no—
189#407C2×D7912yes—
190#343—1584no—
191#324—22no—
192#324—1584no—
193#324—1582no—
194C23×C4—791no—
195C23×C4—791no—
196C23×C4—791no—
197C23×C4—791no—
198C23×C4C4×D7911yescharacteristic
199C23×C4C4×D7911yescharacteristic
200C23×C4C22×D7911yescharacteristic
201C2×M16—22no—
202C2×M16—1582no—
203#125—15816no—
204#125—15816no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—794no—
208C2×C4⋊C4—794no—
209C2×C4⋊C4—1582no—
210C2×C4⋊C4—1582no—
211C2×C4⋊C4—1582no—
212C2×C4⋊C4—1582no—
213C2×C4⋊C4—1582no—
214C2×C4⋊C4—1582no—
215C2×C4⋊C4—1582no—
216C2×C4⋊C4—1582no—
217C2×C4⋊C4—1582no—
218C2×C4⋊C4—1582no—
219C2×C4⋊C4—1582no—
220C2×C4⋊C4—1581no—
221C2×C4⋊C4—1581no—
222C2×C4⋊C4—1581no—
223C2×C4⋊C4—1581no—
224C2×C4⋊C4D31612yes—
225C2×C4⋊C4C22⋊D7912yes—
226C2×C42—1582no—
227C2×C42—1582no—
228C2×C42—1582no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—1582no—
232C22⋊C8—1582no—
233C24C2×C4×D7911yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—3162no—
242C22×C4—3162no—
243C22×C4—3162no—
244C22×C4—3162no—
245C22×C4—792no—
246C22×C4—792no—
247C22×C4—792no—
248C22×C4—792no—
249C22×C4—1581no—
250C22×C4—1581no—
251C22×C4—1581no—
252C22×C4—1581no—
253C22×C4—1581no—
254C22×C4—1581no—
255C22×C4—1581no—
256C22×C4—1581no—
257C22×C4(C2×D79)⋊C412yes—
258C22×C4(C2×D79)⋊C412yes—
259C22×C4D4×D7912yes—
260C22×C4D4×D7912yes—
261C22×C4C2×C22⋊D7911yescharacteristic
262C22×C4C2×D31611yescharacteristic
263M16—44no—
264M16—3164no—
265C2×C8—42no—
266C2×C8—3162no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—1588no—
270C4⋊C4—1588no—
271C4⋊C4—1588no—
272C4⋊C4—1584no—
273C4⋊C4—1584no—
274C4⋊C4—1584no—
275C4⋊C4—1584no—
276C42—1588no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23C22⋊C4×D7911yescharacteristic
281C23C22⋊C4×D7911yescharacteristic
282C23C2×(C2×D79)⋊C411yescharacteristic
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—3162no—
296C2×C4—3162no—
297C2×C4—3162no—
298C2×C4—3162no—
299C2×C4—3162no—
300C2×C4—3162no—
301C2×C4—3162no—
302C2×C4—3162no—
303C2×C4—3162no—
304C2×C4—3162no—
305C2×C4—3162no—
306C2×C4—1582no—
307C2×C4—1582no—
308C2×C4—1582no—
309C2×C4—1582no—
310C2×C4—1584no—
311C2×C4#1457907914yes—
312C2×C4#1457912912yes—
313C2×C4#1457913112yes—
314C8—82no—
315C8—6322no—
316C22—28no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22#1909841811yescharacteristic
324C22#1909853512yes—
325C22C2×#1457907912yes—
326C22#1909854612yes—
327C4—22no—
328C4—22no—
329C4—24no—
330C4—42no—
331C4—3162no—
332C4—3162no—
333C4—3162no—
334C4—3162no—
335C2—22no—
336C2—22no—
337C2—22no—
338C2#2376331611yescharacteristic
339C2#2376332311yescharacteristic
340C2#2376332711yescharacteristic
subgroupG/H
centerC22#19098418
commutator, prime radicalC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1463C2
socleC22×C79#379
cosocleC23×C4×C79C23
hypercenter#1463D79
hypocenterC79#8554
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
176. C79#8554semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145791522
#237634782
#237617302
#237617292
C79×#14632
#855479